Hi there,
The core of your system is the xenyx and computer, connected via USB. Your on-board sound card should be disabled and the xenyx takes its place, handling all input and output.
I'm not sure how useful the Peavey speakers will be as they look more like passive (require an amp) PA speakers.
It doesn't look like your mixer has built in speaker amplification but correct me if I'm wrong there.
Regardless, passive speaker amplifiers or active speakers should have line level inputs via TS/TSR 1/4" or XLR.
Your mixer's main outputs left and right would connect to these inputs and, in your case, the amplifier outputs would connect to the speakers.
The Yamaha piano doesn't appear to have midi or line outputs but it has midi capability over USB and a headphone output.
That means you could either run the phones output to a pair of mixer line inputs, with the appropriate cable (TRS 1/4" to 2x TS 1/4", all male) and record the analog audio as you play,
or you could hook it up via USB and use it to trigger a virtual instrument.
Recording a headphone output isn't really ideal so midi over USB might be the way to go here? It also allows you to make corrections and adjustments very easily after recording, as you're really recording notation data rather than audio.
As for a guitar or two. Electric or electroacoustic direct would want to go through a DI box into a mic input on your mixer.
For a an acoustic recording of acoustic guitar or amp you'd be using a microphone connected to a mic input on the mixer.
One fairly big snag is that your USB mixer is exactly that - It presents only two channels of audio to the computer so if you have, say, 6 microphones plugged in,
you don't get 6 tracks of audio in the computer.
This won't be a problem if you're only intending to record one thing at a time, but it would be very limiting if you were recording a small group or a drum kit, for example.
Hope some of that's useful.
If you've any specific questions, shoot!
Welcome to HR.
Edit: Moving into 'Digital Recording and Computers'.